for \(c=\lbrace c_i\rbrace _{i \in I}\)  having finitely many non-zero terms.
The operator \(G_{\mathcal {A}}(x): =T_{\mathcal {A}}(x)T_{\mathcal {A}}^*(x)\)  on \(\ell ^2(I)\)  is the Gramian of \(\lbrace \psi _i(x): i\in I\rbrace .\)  Let \(\mathcal {A}^{\prime }=\lbrace \varphi _i\rbrace _{i \in I} \subset L^2(X;\mathcal {H})\)  be another countable collection such that \( \mathcal {E}_{\mathcal {D}}(\mathcal {A}^{\prime })\)  is also Bessel, then the operator \(G_{\mathcal {A},\mathcal {A}^{\prime }}(x): = T_{\mathcal {A}}(x)T_{\mathcal {A}^{\prime }}^*(x)\)  is the mixed Gramian of \(\lbrace \psi _i(x)\rbrace _{ i\in I}\)  and \(\mathcal {\lbrace }\varphi _i(x)\rbrace _{i \in I}\)  [1]}. The matrix representations of Gramian and mixed Gramian operator corresponding to the collections \(\mathcal {A}=\lbrace \psi _i: i\in I\rbrace \)  and \(\mathcal {A}^{\prime }=\lbrace \varphi _i: i\in I\rbrace \) , is
\(G_{\mathcal {A}}(x)=\left[\langle \psi _j(x),\psi _i(x)\rangle \right]_{i, j \in I} \ \mbox{and} \ G_{\mathcal {A},\mathcal {A}^{\prime }}(x)=\left[\langle \psi _j(x),\varphi _i(x)\rangle \right]_{i, j \in I}.\) 
